Cyclic groups of automorphisms of schottky type
DOI:
https://doi.org/10.22199/S07160917.1999.0001.00002Abstract
Given an abstract group of orden n, we call its Schottky genus to the minimum genus ? ? 2 on which it acts as group of conformal automorphisms of Schottky type. In this note, we compute the Schottky genus for both cyclic and dihedral groups. In particular, we obtain that the Schottky genus of the dihedral group of order 2n is the same as for the cyclic group of order n. Since every dihedral group is of Schottky type, we have that the Schottky genus of a dihedral group of order 2n is also its minimum genus.
References
[1] L. Bers. Automorphic forms for Schottky groups, Adv. in Math. 16 (1975).
[2] E. Bujalance. Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary, Pacific J. of Math. 109, pp. 279-289, (1983).
[3] W.J. Harvey. Cyclic groups of automorphisms of compact Riemann surfaces, Quart J. Math. Oxford. 17, pp. 86-97, (1966).
[4] R. A. Hidalgo. On Schottky groups with automorphisms, Ann. Acad. Scie. Fenn. Ser. Al Mathematica 19, pp. 259-289, (1994).
[5] R. A. Hidalgo. Schottky uniformizations of closed Riemann surfaces with abelian groups of conformal automorphisms, Glasgow Math. J. 36, pp. 17-32, (1994).
[6] R. A. Hidalgo. Dihedral groups of Schottky type. Preprint.
[7] R. A. Hidalgo. A4, A5 and S4 of Schottky type. Preprint.
[8] F. Klein. Neue Beiträge zur Riemann'schen Funktiontheorie, Math. Ann. 21, pp. 141-218, (1883).
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[11] B. Maskit. Kleinian groups. G.M.W. 287, Springer Verlag, (1988).
[2] E. Bujalance. Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary, Pacific J. of Math. 109, pp. 279-289, (1983).
[3] W.J. Harvey. Cyclic groups of automorphisms of compact Riemann surfaces, Quart J. Math. Oxford. 17, pp. 86-97, (1966).
[4] R. A. Hidalgo. On Schottky groups with automorphisms, Ann. Acad. Scie. Fenn. Ser. Al Mathematica 19, pp. 259-289, (1994).
[5] R. A. Hidalgo. Schottky uniformizations of closed Riemann surfaces with abelian groups of conformal automorphisms, Glasgow Math. J. 36, pp. 17-32, (1994).
[6] R. A. Hidalgo. Dihedral groups of Schottky type. Preprint.
[7] R. A. Hidalgo. A4, A5 and S4 of Schottky type. Preprint.
[8] F. Klein. Neue Beiträge zur Riemann'schen Funktiontheorie, Math. Ann. 21, pp. 141-218, (1883).
[9] P. Koebe. Über die Uniformisierung der Algebraischen Kurven II, Math. Ann. 69, pp. 1-81, (1910).
[10] C. Maclachlan. Abelian groups of automorphisms of compact Riemann surfaces, Proc.LOndon Math. Soc. 15, pp. 600-712, (1965).
[11] B. Maskit. Kleinian groups. G.M.W. 287, Springer Verlag, (1988).
Published
2018-04-04
How to Cite
[1]
R. A. Hidalgo, “Cyclic groups of automorphisms of schottky type”, Proyecciones (Antofagasta, On line), vol. 18, no. 1, pp. 13-21, Apr. 2018.
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